Find the Nullity [[1,-1,-2],[-1,-1,-3+k]]

Problem

Nullity*[[1,−1,−2],[−1,−1,−3+k]]

Solution

1. Set up the matrix and identify its dimensions. The matrix A is a 2×3 matrix.

A=[[1,−1,−2],[−1,−1,−3+k]]

2. Perform row reduction to find the Row Echelon Form. Add the first row to the second row.

(R_2)+(R_1)⇒(R_2)

[[1,−1,−2],[0,−2,−5+k]]

3. Determine the rank of the matrix. The rank is the number of non-zero rows in the row-reduced form. Since the first two columns of the reduced matrix are linearly independent regardless of the value of k the rank is 2.

rank(A)=2

4. Apply the Rank-Nullity Theorem, which states that for an m×n matrix, the rank plus the nullity equals the number of columns n

rank(A)+nullity(A)=n

5. Substitute the values into the theorem. Here, n=3 and rank(A)=2

2+nullity(A)=3

nullity(A)=3−2

nullity(A)=1

Final Answer

nullity*[[1,−1,−2],[−1,−1,−3+k]]=1

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